The key assumptions in linear programming include the following: first, the relationships between decision variables are linear, meaning both the objective function and constraints can be expressed as linear equations. Second, the principle of additivity holds, indicating that total contributions are the sum of individual contributions. Third, divisibility is assumed, allowing decision variables to take any real values, including fractions. Fourth, certainty is assumed, meaning all coefficients in the objective function and constraints are known and constant. Finally, non-negativity is required, stating that decision variables must be zero or positive, reflecting real-world constraints.